Crosstalk cancellation for a simultaneous phase shifting interferometer

1. A method of minimizing fringe print-through in a phase-shifting interferometer, comprising the steps of:

(a) determining multiple transfer functions of pixels in the phase-shifting interferometer;

(b) computing a crosstalk term for each transfer function, and

(c) displaying, to a user, a phase-difference map using the crosstalk terms computed in step (b),

wherein computing crosstalk terms in step (b) includes computing an N-dimensional vector, where N corresponds to the number of transfer functions, and

the N-dimensional vector is obtained by minimizing a variance of a modulation function, and

wherein N has a value of four, and

the modulation function, defined as gamma, includes the following relationship:

gamma=2*sqrt(( d 1 −n 1Cancel)^2+( n 1 −d 1Cancel)^2))/intensity;

in which:

n1=y1−y3;

d1=y4−y2;

d1Cancel=a(1)*y2+a(2)*y4;

where d1Cancel is a cancellation term, and

a is a vector of the coefficients, y1, y2, y3 and y4;

n 1Cancel= a (3)* y 1 +a (4)* y 3;

where n1Cancel is a cancellation term, and

a is a vector of the coefficients, y1, y2, y3 and y4;

intensity= y 1 +y 2 +y 3 +y 4; and

y1, y2, and y4 are four interferogram images.

2. The method of claim 1 wherein the phase-shifting interferometer includes a reference beam and a test beam, and

determining a transfer function in step (a) includes

measuring intensities of the reference beam and the test beam at each pixel, and

measuring an optical path difference between the reference beam and the test beam at each pixel.

3. The method of claim 2 wherein determining the transfer function includes solving the following mathematical relationship:

F ( x,y )=½( I r +I s +2√{square root over ( I r I s )} cos(Δφ( x,y )))

where F is a transfer function at pixel location (x,y);

I r (x, y) and I s (x, y) are intensities of reference and test wavefronts of the phase shifting interferometer, at each (x,y) coordinate in the image, respectively; and

Δφ(x,y) is an optical path difference between the reference and test wavefronts.

4. The method of claim 2 wherein

each pixel includes pixels along a diagonal in an image array of the phase-shifting interferometer.

5. The method of claim 1 wherein determining multiple transfer functions in step (a) includes obtaining N interferograms, where N corresponds to N transfer functions, and

each interferogram has a total number of pixels equal to (n×m)/N,

where n and m are the number of pixels in a detector array of the phase-shifting interferometer.

6. The method of claim 1 wherein displaying the phase difference-map is based on the following expression:

phase=arctangent[( d 1 −n 1Cancel)/( n 1 −d 1Cancel)].

7. The method of claim 1 further including the following step:

(d) combining pixels having similar transfer functions to generate a single image per transfer function.

8. A processor containing a set of instructions for minimizing fringe print-through in a phase-shifting interferometer, The instructions comprising the steps of:

(a) determining multiple transfer functions of pixels in the phase-shifting interferometer;

(b) computing a crosstalk term for each transfer function, and

(c) displaying, to a user, a phase-difference map using the crosstalk terms computed in step (b),

wherein a modulation image, defined as gamma, includes the following relationship:

gamma=2*sqrt(( d 1− n 1Cancel)^2+( n 1− d 1Cancel)^2))/intensity;

in which:

n1=y1−y3;

d1=y4−y2;

d1Cancel=a(1)*y2+a(2)*y4;

where d1Cancel is a cancellation term, and

a is a vector of the coefficients, y1, y2, y3 and y4;

n 1Cancel= a (3)* y 1 +a (4)* y 3;

where n1Cancel is a cancellation term, and

a is a vector of the coefficients, y1, y2, y3 and y4;

intensity= y 1+ y 2 +y 3 +y 4; and

y1, y2, y3 and y4 are four phase shifted images.

9. The processor of claim 8 including the following steps:

(d) forming a modulation function that includes the transfer functions computed in step (a); and

(e) minimizing a variance of the modulation function.

10. The processor of claim 9 wherein

the crosstalk term for each transfer function includes a value multiplied by at least one transfer function determined in step (a).

11. The processor of claim 8 wherein the phase-shifting interferometer includes a reference beam and a test beam, and

determining a transfer function in step (a) includes

measuring intensities of the reference beam and the test beam at each pixel, and

measuring an optical path difference between the reference beam and the test beam at each pixel.

12. The processor of claim 8 wherein

determining multiple transfer functions of pixels includes:

measuring intensities of pixels of an image array in the phase-shifting interferometer.

13. The processor of claim 8 wherein

determining multiple transfer functions of pixels includes:

measuring intensities of pixels along a diagonal of an image array in the phase-shifting interferometer.